A kinetic model for the transport of electrons in a graphen layer

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1 A kinetic model for the transport of electrons in a graphen layer Clotilde Fermanian Kammerer, Florian Méhats To cite this version: Clotilde Fermanian Kammerer, Florian Méhats. A kinetic model for the transport of electrons in a graphen layer <hal v> HAL Id: hal Submitted on 9 Jun 015 v, last revised 7 Sep 017 v3 HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

2 A KINETIC MODEL FOR THE TRANSPORT OF ELECTRONS IN A GRAPHEN LAYER CLOTILDE FERMANIAN KAMMERER AND FLORIAN MÉHATS Abstract. In this article, a kinetic model for the transport of electrons in graphene is derived with the tools of semiclassical analysis. The underlying quantum model is a massless Dirac equation, whose eigenvalues display a conical singularity responsible for non adiabatic transitions between the two modes. Our kinetic model takes the form of two Boltzmann equations coupled by a collision operator modeling these transitions. This collision term includes a Landau-Zener transfer term and a jump operator whose presence is essential in order to ensure a good energy conservation during the transitions. We propose an algorithmic realization of the semi-group solving the kinetic model, by a particle method. In the last section, a series of numerical experiments are given in order to study the influences of the various sources of errors between the quantum and the kinetic models. 1. Introduction 1.1. Graphene structures. Recently, graphene based structures have been the object of intensive research in nanoelectronics, see for instance the reviews [5, 10] and references therein. Graphene is a single D sheet of carbon atoms in a honeycomb lattice and, differently from conventional semiconductors, the most important aspect of graphene s energy dispersion is its linear energy-momentum relationship. Electrons behave as massless relativistic particles, the conduction and valence bands intersecting at the zero energy point, with no energy gap. These features enable to observe at low energy some physical phenomena of quantum electrodynamics, such as Klein tunneling that is, the fact that Dirac fermions can be transmitted through a classically forbidden region. We are here interested in the description of the transport of electrons in a graphene device via a kinetic model. Kinetic models are usually easier to implement numerically and have a cheaper numerical cost, compared to out-of-equilibrium full quantum models. Moreover, the treatment of boundary conditions is simpler in this framework, which also enables to enrich the description by adding collisional effects via Boltzmann-like terms. However, due to the absence of gap between the conduction and valence bands, it is not correct to describe separately electrons and holes, which remain coupled even at the semiclassical limit. The objective of this paper is to introduce a kinetic model for ballistic transport, which treats the possible transitions between bands. This kinetic model is derived rigorously in a linear setting and leads to algorithmic realizations which is tested numerically. Similar The authors would like to express their gratitude to Caroline Lasser for her help. This work was supported by the ANR-FWF Project Lodiquas ANR-11-IS and by the ANR project Moonrise ANR-14-CE

3 C. FERMANIAN KAMMERER AND F. MÉHATS strategy has been developed at the same moment where we were writing this paper by A. Faraj and S. Jin in [1]. 1.. The quantum model. The kinetic model that will be introduced below consists in a system of approximate equations based on the Wigner counterpart of an underlying quantum transport model. At the quantum level, the ensemble of particles is described by its density matrix ϱt, solving the von Neumann equation The Hamiltonian reads H = i v F σ X + eu = v F i T ϱ = [H, ϱ]. 0 i X1 X i X1 + X 0 + eu, where X R, v F is the Fermi velocity, σ = σ X1, σ X denotes the Pauli matrices vector and U = UX is a potential, see [10] for physical references. Let us first put this equation in dimensionless form. We introduce a characteristic space length L, a characteristic energy E and a characteristic density n, then define the associated characteristic time by t = and denote E x = X L, t = T t, V = eu E, ϱ = ϱ nl. The system in dimensionless form reads 1.1 i t ϱ = [AD + V, ϱ ], where the semiclassical dimensionless parameter is where D = i x and A is the matrix Aξ = = v F EL 1, 0 ξ1 iξ ξ 1 + iξ 0 The matrix Aξ has two eigenvalues ξ and ξ with associated eigenprojectors Π + and Π, Π ± = 1 Id ± 1 ξ Aξ where Id is the identity matrix. The singularity of the eigenvalues at the point ξ = 0 is called conical singularity. The applied potential V x is supposed to be smooth. Finally, we assume that for any > 0, the initial data ϱ 0 is simultaneously a nonnegative trace-class operator and a Hilbert-Schmidt one. We shall denote by L 1 L R the set of trace-class operators on L R and by L L R the space of the Hilbert-Schmidt operators. We shall assume that the family of operator ϱ 0 >0 is a bounded family of L 1 L R d, that is. 1. C > 0, ϱ 0 L 1 L R C. We shall not assume any bound uniform in on the Hilbert-Schmidt norm of ϱ 0. Note that under these assumptions, we obtain t R, > 0, ϱ t L 1 L R L L R,

4 KINETIC MODEL FOR GRAPHEN 3 with t R, > 0, ϱ t L 1 L R C Wigner functions. Denoting now by ρ t, x, y the integral kernel of ϱ, the Wigner function is defined by w t, x, ξ = 1 π e iξ η ρ t, x η, x + η dη. Since ρ t is Hilbert-Schmidt, its kernel is a function of L R x R y and similarly for w t. Note however that this fact holds for any > 0 without any uniform bound. The fact that the family ρ t >0 is bounded in L 1 L R implies that the family of distributions w t >0 is bounded in the set of distributions see Remark.9 We call diagonal part of the Wigner transform the scalar distributions 1.3 w ±t, x, ξ = tr Π ± ξw t, x, ξπ ± ξ and we have w t, x, ξ = w +t, x, ξπ + ξ+w t, x, ξπ ξ+π + w t, x, ξπ +Π w t, x, ξπ +. When is small, the off-diagonal contribution to the Wigner transform is known to be highly oscillating in time so that Π ± w t, x, ξπ 0 0 in D R t R d \ {ξ = 0}, see []. For this reason, we focus on the quasi-distribution functions w±t, x, ξ. Far from the crossing set {ξ = 0}, w+ and w satisfy approximated transport equations 1.4 t w + + ξ ξ xw + V x ξ w + = O, t w ξ ξ xw V x ξ w = O, in D R t R d \ {ξ = 0}. Besides, the equations 1.4 imply that, outside {ξ = 0}, the functions w± are constant along the integral curves Φ t ± t R of the vector fields 1.5 H ± x, ξ = ± ξ ξ x x V ξ. Such curves also called Hamiltonian curves of ± ξ + V x are well-defined and smooth as long as they do not reach {ξ = 0}. They satisfy Φ t ±x, ξ = H ± Φ t ± x, ξ, Φ 0 ± = x, ξ. It is proved in [14] see Proposition 3 therein that such curves may reach {ξ = 0} in finite time, and that, if V 0 at the impact point, the curve can be prolongated in a unique way away from {ξ = 0} generating a continuous trajectory which is not C 1. These facts are recalled in details in Section.1.1 below. It is also important to notice that the evolution of w + and w are decoupled at leading order outside {ξ = 0}: this regime is said to be adiabatic. The singularity of the eigenvalues of Aξ when ξ = 0 is known to produce non adiabatic transitions between the modes. Our aim here is to propose a kinetic model which is also valid

5 4 C. FERMANIAN KAMMERER AND F. MÉHATS close to ξ = 0. We are going to add a collision kernel to these equations which will couple the evolutions and generate transitions between the modes Conical singularities. Systems presenting conical singularities have been the subject of extensive works since the early thirties with the works of Landau and Zener [7, 33]. Such singularities arise in particular when studying molecular dynamics in the frame of Born-Oppenheimer approximation see [31, 8] for example. Pioneer works have been performed in this context by G. Hagedorn and his collaborators, with a wave-packet approach [3, 4]. Several ideas used here are due to these contributions. Ten years ago, classification of crossings for rather general systems was performed independently by [6, 7] and [15]. In the latter reference and in [14], the analysis of the crossing is made from the point of view of Wigner transform and can be adapted to our setting. This kind of analysis has led to numerical realizations for molecular propagation [9], [17] and [18] and we have been inspired by these results. Of course, the Dirac equation arising in the graphen context presents major difference, when compared to the Schrödinger equation which models molecular propagation. However, the transitions due to the conical intersections can be treated similarly. The collision kernel which treats the transitions arising from the conical intersections, is derived from the analysis of conical intersections performed in [14] and from the particle description derived in [9, 17, 18, 19] for molecular dynamics. Precise statements are given below The approximate kinetic model. We consider the set Σ defined by Σ = {x, ξ R 4, ξ V x = 0} which is an hypersurface of R 4 under the assumption 1.6 V x 0 x R. This set is the place where the gap between the two modes i.e. the function ξ is minimal along the trajectories see Remark 1.5. We notice that, assuming 1.6, the vector fields H ± x, ξ defined in 1.5 are transverse to Σ in a neighborhood of {ξ = 0}. This comes from the observation that 1.7 V x H ± ξ + d V xξ H ± x = V x ± d V xξ ξ ξ < 0 if ξ is small enough. As a consequence, when the trajectories reach their minimal distance to the gap, they pass through Σ, arriving from the region {ξ V x > 0} and going to the region {ξ V x < 0}. We define f+t, f t as a pair of solutions to the following system: t f+ + ξ ξ xf+ V x ξ f+ = K + f+, f 1.8 t f ξ ξ xf V x ξ f = K f+, f with initial conditions f +0 = w +0 and f 0 = w 0 and where K ± are two collision kernels, defined below in 1.11 and 1.1. The collision process is involved above Σ; as a consequence, outside Σ, the functions f ± are constant along the curves Φ t ± t R introduced previously. Starting from a initial data localized far from Σ, the solution f ± of system 1.8 is obtained

6 KINETIC MODEL FOR GRAPHEN 5 by propagating the data by the flow Φ t ± t R so that the plus and the minus modes have decoupled evolutions. Whenever trajectories reach Σ, the transition kernel will generate transfers between the modes. Even for smooth initial data, the result of this process will not be smooth functions and they will present discontinuities on Σ. But we have to localize on Σ functions that present discontinuities through it, and thus have different traces. In order to distinguish two sides of Σ, we will take advantage from the fact that, as noticed above, the flows H ± are transverse to Σ in suitably chosen neighborhoods Ω of points x 0, ξ 0 such that V x 0 0 and ξ 0 is small enough. For a function g which is defined in such Ω, we set for x, ξ Ω Σ 1.9 g Σ,in x, ξ = lim s 0 g Φ s ±x, ξ and g Σ,out x, ξ = lim s 0 + g Φ s ±x, ξ. Let us now describe these collision kernels K ±. They depend on a transfer coefficient 1.10 T x, ξ = exp π ξ, V x and on two jump operators V x J ± x, ξ = x ± ξ V x, ξ, x, ξ Σ. Then, for functions f C 1 R 4 \ Σ, which have traces on Σ, we distinguish between entering and outgoing traces by denoting by f Σ,in the restriction to Σ of the function f 1 {ξ V x 0} and by f Σ,out the restriction to Σ of f 1 {ξ V x 0}.This is adapted to functions which are discontinuous through Σ. Then, the collision kernels K ± are defined by K + f, g = Λ + x, ξδ Σ x, ξ T f Σ,in T g Σ,in J +, K f, g = Λ x, ξδ Σ x, ξ T g Σ,in T f Σ,in J, where the Jacobians Λ ± are defined by 1.13 x, ξ Σ, Λ ± x, ξ = V x ± ξ 1 d V xξ ξ V x + d V xξ. Remark 1.1. Note that the transfer coefficient T x, ξ is exponentially small as soon as ξ > R for some R > 0. Moreover, if ξ R, we have J ± = Id + OR and Λ ± x, ξ = V x + OR, under the assumption 1.6. Since the initial data that we want to treat are not necessarily continuous outside Σ, something has to be said in order to extend the collision kernel to locally integrable functions. For this and for stating the main result, we shall need a certain number of assumptions. Assumption The initial data ϱ 0 >0 satisfies 1. and its Wigner transform w 0 >0 is localized away from Σ. The potential is non degenerated: V x 0.

7 6 C. FERMANIAN KAMMERER AND F. MÉHATS 3 We have w 0 = O 1/8 in L 1 R d and the symbol a and the time T are such that within the time interval [0, T ], each of the trajectories Φ t + arriving at the support of a at time T has passed through Σ at most once. Our main result is the following theorem, which states that the functions f ± provides an approximation of the Wigner transforms w ±. Theorem 1.3. If 1 and of Assumption 1. is satisfied on the time interval [0, T ], then 1.8 admits a unique weak solution. Moreover if a C0 R is compactly supported outside {ξ = 0} and χ Cc [0, T ], R satisfy 3 of Assumption 1., then there exist positive constants C, 0 > 0 such that, for all 0 < < 0, 1.14 R tr χt f± w± t, x, ξ ax, ξ dx dξ dt C 1/8. 5 Remark The existence of solutions to these kinetic equations comes from the fact that, under 1 and of Assumption 1., these equations have a particle description relying on a Markov semi-group that is explained in the next section and is crucial for the proof of the theorem. Note that the role of the indexes plus and minus can be inverted in 3 of Assumptions 1. and result of Theorem 1.3 still holds. 3 The limitation induced by 3 to the range of validity of Theorem 1.3 comes from the fact that the kinetic kernels K ± are not adapted in some situations where the modes interfere too much. It appears nevertheless that these kernels description encounter a larger range of situation than those satisfying 3, as it appears in the numerical realizations of Section 3. Some example of situation where 3 is not satisfied and where the description by the kernels K ± fails is given in [0] in the context of conical intersections for molecular dynamics. 4 We refer to Remark 3.1 below where additional comments are performed in view of the semi-group description of w±t A Markov semi-group. We consider the Hamiltonian flows Φ t + = x t +x, ξ, ξ t +x, ξ and Φ t = x t x, ξ, ξ t x, ξ with x 0 ±x, ξ, ξ±x, 0 ξ = x, ξ and d dt xt + = ξt + ξ t and d dt ξt + = V x t +, d dt xt = ξt ξ t and d dt ξt = V x t. As long as ξ 0, the smoothness of the Hamiltonians ξ ± V x yields local existence and uniqueness of the trajectory passing through x, ξ for any x R. However, it may happens that ξ t ± t t 0 for some t R and some index + or. If at the point x t ±, the assumption 1.6 is satisfied that is if V x t ± 0, then one can prove that there exists a unique continuation to the map t Φ t ± when t > t see Proposition 1 in [15] and Proposition.1 below where a precise statement and a proof are given for the convenience of the reader. As a consequence, the assumption 1.6 guarantees the existence and uniqueness of the solutions to However, these trajectories are no longer smooth when passing through ξ = 0; more

8 KINETIC MODEL FOR GRAPHEN 7 precisely, the vector Φ t has a discontinuity at t = t whenever ξ t ± = 0. It is also interesting to notice that if the latter assumption 1.6 fails at x t ±, then uniqueness is no longer guaranteed. Remark 1.5. For any trajectory Φ t ±, the quantity ξ t ± reaches its minimum on Σ. Indeed, we can check that d ξ t dt ± = ξ± t V x t ±, which is equal to 0 whenever Φ t ± Σ. This motivates the introduction of the set Σ. We are now going to introduce a branching process between both types of trajectories. We attach the labels + and to the phase space and for points x, ξ, j R 4 ± := R 4 {+1, 1}, we consider trajectories T x,ξ,j : [0, + R 4 ±, which combine deterministic classical transport and random jumps between the levels at the manifold Σ. More precisely, we set T x,ξ,j t = Φ t jx, ξ, j as long as Φ t j x, ξ Σ. Whenever the deterministic flow Φt j x, ξ hits the manifold Σ at a point x, ξ, a random jump from occurs with probability T x, ξ. x, ξ, j to J j x, ξ, j The jump aims at preserving at order O the energy of the trajectory for points ξ where the transfer coefficient is relevant, that is points where ξ R according to Remark 1.1. This is an important ingredient of the proof see Remark.13. Indeed, set E ± x, ξ := ± ξ + V x, then, if ξ R, at a jump from + to, we have E J + x, ξ = E + x, ξ + O ξ = E + x, ξ + OR. Note that the importance of the jump has been illustrated numerically in [19] in the context of molecular propagation ; this jump was already performed in [4] for the construction of gaussian wave packets which are approximated solutions of a Schrödinger equation with matrix valued potential presenting a conical intersection. Since V x 0, the trajectories which reach the manifold Σ arrive there transversally to Σ. As a consequence, in each bounded time interval [0, T ], each path x, ξ, j T x,ξ,j t has a finite number of jumps and remains in a bounded region of the phase space R 4 ±. Besides, away from the jump manifold Σ { 1, +1} each path is smooth. Following [9, 30], we define the function P x, ξ, j; t, as the function which associate to a measurable set Γ R 4 ± the probability P x, ξ, j; t, Γ of being at

9 8 C. FERMANIAN KAMMERER AND F. MÉHATS time t in Γ having started in x, ξ, j. And we define a time-dependent Markov process L t t 0 acting on bounded measurable scalar functions f : R 4 ± C by L t fx, ξ, j := fq, p, k P x, ξ, j; t, dq, p, k. R 4 { 1,+1} An explicit expression of L t is written on short interval times close to jump points in Section.. see equations.7 and.8 below. In order to define its action on Wigner functions, we need to identify pairs of functions a +, a with some function a on R 4 ±, which is done by the identification 1.16 ax, ξ, ±1 = a ± x, ξ, x, ξ R 4. Through this identification, the action of L t t 0 on C 0 R 4 \ {ξ = 0} is given by L t ax, ξ := L t a x, ξ, +1, L t a x, ξ, 1, We extend this action to D R 4 \ {ξ = 0} by duality by setting f D R 4, a C 0 R 4 \ {ξ = 0}, L t f±1, a = f, L t a±1. Proposition 1.6 Resolution of the kinetic model. Set w 0 = w +0, w 0 and assume that 1 and of Assumptions 1. are satisfied. Then the function f t, x, ξ = L t w 0x, ξ is the unique solution to system 1.8 in D R 4 \ {ξ = 0}. Then, Theorem 1.3 is a corollary of the following proposition. Proposition 1.7 Approximation by the semi-group. Set w 0 = w+0, w 0 and assume for χ Cc [0, T ], R and a C0 R 4 \ {ξ = 0}, Assumptions 1. are satisfied, then, there exist positive constants C, 0 > 0 such that for all 0 < < 0, R tr χt w±t, x,, ξ L t w 0x, ξ, ±1 ax, ξ dx dξ dt C 1/8. d+1 Remark 1.8. Note that the hypothesis 3 of Assumptions 1. imply that on the interval [0, T ], the trajectories which reach the support of a has performed at most one jump. This Markov semi-group suits to an algorithmic resolution which is developed in Section 3.1, where we provide some numerical experiments which testify to the performance of the algorithm.. Proofs In this section, we prove the main result, Theorem 1.3. After a preliminary section devoted to the study of the classical flow, we proceed in two steps. We first prove Proposition 1.6 which relates the approximated system 1.8 and the Markov semi-group described in Section 1.6. Then we give a sketch of the proof of Proposition 1.7, which explains why this semi-group describes the evolution of the Wigner transform of the solutions of 1.1; for this, we follow the proof of Theorem. in [17] which has been performed for a system of Schrödinger equations coupled by a matrix-valued potential..1. Preliminaries. In this section, we analyze precisely the geometry close to a point of Σ in order to precise the setting in which the proofs will be performed.

10 KINETIC MODEL FOR GRAPHEN The generalized flow. In this section, we gather some properties of the flows Φ t ± that will be useful in the next sections. We first focus on the existence and uniqueness of the generalized trajectories of the Hamiltonian vector fields H ± and recall the arguments of the proof given in [14]. Proposition.1. For any x 0 R such that V x 0 0, there exists τ 0 > 0 and a unique Lipschitz continuous map t x t ±x 0, 0, ξ t ±x 0, 0, t [ τ 0, τ 0 ] satisfying 1.15 for t 0 and such that x 0 ±x 0, 0 = x 0, ξ 0 ±x 0, 0 = 0 and ẋ t ±x 0, 0 t 0 V x 0 V x 0, ξt ± x 0, ξ 0 t 0 V 0 x 0. Corollary.. With the notations of Proposition.1, we have lim H + x t ±, ξ± t = lim H x t ±, ξ± t = V x 0 ξ + V x 0 t 0 t 0 + V x 0 x, lim H x t ±, ξ± t = lim H x t ±, ξ± t = V x 0 ξ V x 0 t 0 + t 0 V x 0 x. In the following and with the notations of Proposition.1, we shall set.1 Hx 0 := lim +x t ±, ξ± t = lim x t ±, ξ±, t t 0 t 0 +. H x 0 := lim x t ±, ξ± t = lim x t ±, ξ±. t t 0 + t 0 We observe that if ω = dξ dx is the canonical skew-symmetric -form of the cotangent space of R, we have.3 ωh, H = V x 0 > 0. Proof of Proposition.1. Following Proposition 3 in [14], we introduce two flows Ψ t 1 = x t 1, ξ1 t and Ψ t = x t, ξ t which are defined by { ẋ t j = 1j+1 sgnt ξt j ξj t, x0 j = x 0, ξ j t = V xt j, or equivalently ξ t j = t 1 0 t V x ts j ds, x t j = x j V xsσ j dσ ds. V xsσ j dσ The last system can be solved on short time by a fixed point argument in an open subset of { V x 0} and the resulting map x Ψ t j x, 0 is smooth. As a consequence, there exists a neighborhood Ω of x 0, 0 such that Ω { V x 0} and τ 0 > 0 such that for all x, 0 Ω, the maps Φ t + x, 0 t [0,τ = 0] Ψ t 1x, 0 t [0,τ 0] and Φ t +x, 0 t [ τ = 0,0] Ψ t x, 0 t [ τ, 0,0] Φ t x, 0 t [0,τ 0] = Ψ t x, 0 t [0,τ 0] and Φ t x, 0 t [ τ 0,0] = Ψ t 1x, 0 t [ τ 0,0], solve our problem. The flows Φ t ± which are well defined for ξ 0 extend to Lipschitz continuous maps t Φ t ±x, ξ, t [ τ 0, τ 0 ], x, ξ Ω.

11 10 C. FERMANIAN KAMMERER AND F. MÉHATS Remark.3. Following the arguments of Section 6. in [16], one can prove that for t < τ 0 and α N d, the maps x, ξ α x Φ t ±x, ξ are continuous maps on Ω with bounded locally integrable time derivatives t α x Φ t x, ξ..1.. Local analysis. In what follows, we shall associate with points x, ξ, which are close enough to the set Σ, a number τ ± x, ξ which is the time that separates x, ξ from the point of the trajectory Φ t ±x, ξ which belongs to Σ. Proposition.4. Let x 0 R such that V x 0 0, there exists an open set Ω { V x 0} R 4 x,ξ containing x 0, 0 and such that the relations Φ τ±x,ξ x, ξ Σ define two continuous functions on Ω x, ξ τ ± x, ξ. Remark.5. By definition of τ ± x, ξ, we have V x ± τ ±x,ξ ξ± τ ±x,ξ = 0 for any x, ξ Ω. Proof. Let us study the plus mode. We observe that.4 d dt ξt + = V x t + ξ+ t and d dt ξt + = V x t + d V x t +ξ+ t ξ+ t ξ+ t Since V x 0 0, we can find a neighborhood U of x 0, 0, c 0 > 0 and τ 1 > 0 such that d.5 x, ξ Ω 1, t τ 1, dt ξt +x, ξ c 0 > 0. Because of.5, the map t ξ+ t reaches its minimum at most once in U. With any x, ξ U, we associate an interval [t i x, ξ, t f x, ξ] of maximal size such that Φ t x, ξ U for all t ]t i x, ξ, t f x, ξ[. Because of the first relation of.4, we are interested to the times where the curves Φ t x, ξ crosses the hypersurface Σ, which happens at most once in U. We set Ω + = { x, ξ U, y, η Σ U, s ]t i y, η, t f y, η[, x, ξ = Φ s +y, η }. Then Ω + is a neighborhood of x 0, 0 included in U and such that the relation Φ τ+x,ξ + x, ξ Σ defines a map τ + from Ω + into R. We define similarly Ω and the map τ x, ξ and we choose Ω = Ω + Ω see Fig. 1. It is classical to prove that τ + is a continuous map on Ω. We set z = x, ξ and φt, z = d dt ξ+ t = V x t +z ξ t +z and we argue by contradiction. We assume that there exist α 0, z 0 and a sequence z n n N going to z 0 as n goes to + and such that τ + z n τ + z 0 > α 0. Then, by the continuity of φ, we get φτ + z 0, z n n + 0 and since tφ c 0 by.5, we have c 0 τ + z 0 τ + z n φτ + z 0, z n φτ + z n, z n = φτ + z 0, z n. Taking n large enough, we get τ + z n τ + z 0 < α 0 /, whence a contradiction and the continuity of the map z τ + z is proved.

12 KINETIC MODEL FOR GRAPHEN 11 Σ Φ t Ω Ω + Φ t + Ω + Ω in Ω out x 0, 0 Φ t + Ω Φ t Figure 1: Domains Ω + blue+green, Ω yellow+green and Ω green = Ω + Ω = Ω in Ω out One argues similarly for the mode minus. Remark.6. The hypersurface Σ parts Ω into two distinct connected regions see Fig. 1:.6 Ω out := {τ + < 0, τ < 0} Ω = { V x ξ < 0} Ω Ω in := {τ + > 0, τ > 0} Ω = { V x ξ > 0} Ω. Indeed, if V x ξ > 0, we have simultaneously, d dt ξt + < 0 and d dt ξt < 0, thus we have τ + x, ξ > 0 and τ x, ξ > 0. Besides, for x, ξ Ω in, we have Φ t x, ξ Ω in for t [0, τ + x, ξ[. This remark is fundamental to prove that the kinetic system 1.8 has a meaning in Lebesgue space L 1 R 4 and then to solve it. Indeed, the collision kernel requires to take the trace of the functions on Σ, which is not necessarily doable for L 1 - functions. We consider the open set Ω of Proposition.4 where H ± are transverse to Σ. We introduce the sets Ω in and Ω out defined by.6. We can find an open set V, V Ω such that there exists t i < t f i for initial and f for final with and we define see Fig.. y, η Σ V, Φ ti ±y, η Ω in, y, η Σ V, Φ t f ± y, η Ω out, Σ i ± = {Φ ti ±y, η, y, η Σ V} Ω in, Σ f ± = {Φ t f ± y, η, y, η Σ V} Ω out, Lemma.7. With the above notations, let f ± be two functions which are invariant by Φ s ± in Ω in V = { V x ξ > 0} V. Then f ± have traces on Σ V that we

13 1 C. FERMANIAN KAMMERER AND F. MÉHATS Σ Ω Σ i + Σ V Σ f + Figure : Surfaces Σ i +, Σ f + and Σ V denote by f ± Σ,in. Similarly, if f ± are invariant by Φ s ± in Ω out V = { V x ξ < 0} V. Then f ± have traces on Σ V that we denote by f ± Σ,out. Proof. We first use the functions τ ± for defining the four following continuous maps and κ i ± : Σ i ± Σ, x, ξ Φ τ±x,ξ ± x, ξ, κ f ± : Σ f ± Σ, These four maps are homeomorphisms. Set x, ξ Φ τ±x,ξ ± x, ξ. f ± Σ,in = κ i ± f ±t i. These two functions are L 1 functions of Σ Ω which are the traces of f ± on Σ. We argue similarly in Ω out... Analysis of the kinetic equations 1.8. In this section we prove the existence and uniqueness of solutions to the kinetic equations 1.8 and we analyze the link between the Markov semi-group and this system of equations...1. Uniqueness of solutions to 1.8. As a corollary of the analysis of the previous subsection, we obtain that the solutions of 1.8 are unique if they do exist. Indeed, by 1 of Assumptions 1., the data has been chosen supported outside Σ so that for short time the kinetic system 1.8 reduces to classical transport by the two flows. We cut the data in a sum of compactly supported pieces so that each of these pieces satisfy the hypothesis of Lemma.7 after a certain amount of time. We only need to consider one of these pieces and we take the notations of Lemma.7. Because of the previous decomposition of the data, we may assume that f ±t i is compactly

14 KINETIC MODEL FOR GRAPHEN 13 supported in V Ω in. By Lemma.7, if the solution does exist on the interval of time [t i, t f ], we must have f+t, x, ξ = 1 τ+>0fσ,in Φ τ+x,ξ x, ξ + 1 τ+<0fσ,out Φ τ+x,ξ x, ξ. As a consequence, the solution for the plus mode will be unique if it exists if and only if the outgoing trace f + Σ,out is uniquely determined by the entering traces f + Σ, and f Σ,+. We shall use the following Lemma, the proof of which is postponed to the end of the section. We gather in this Lemma properties that we will use in the following and we postpone their proof at the end of the section. Lemma.8. In the set of distributions, we have t + H ± τ ± x, ξ + t = 0, t + H ± Φ τ±x,ξ ± x, ξ = 0, H + 1τ+x,ξ<0 = Λ+ x, ξδ Σ x, ξ, where Λ + is given by As a consequence of this Lemma, we also have in D Ω out, H + τ + = 1, which implies H + Φ τ+x,ξ x, ξ = H + τ + Φτ +x,ξ x, ξ + H + Φ τ+x,ξ x, ξ = 0. Therefore, the fact that implies As a consequence, t + H + f + = K + f + Σ,in, f Σ,out, Λ + f + Σ,in f + Σ,out = Λ+ T f + Σ,in T f Σ,in J +. f + Σ,out = 1 T f + Σ,in T f Σ,in J +, and the outgoing and ingoing traces are linked. A similar argument holds for the minus mode and as a consequence, the solution of 1.8 is unique if it exists.... Link between the particle description and the kinetic model. Let us prove Proposition 1.6. We aim at proving that the semigroup L t provides the unique solution of 1.8. In fact, since the uniqueness property has been proved at the end of Subsection..1, we only have to prove that if g t = L t w 0, t R +, and g ±t, x, ξ = g t, x, ξ, ±1, then g +, g satisfies system 1.8 in D R 4 \ {ξ = 0}. By density of compactly supported continuous functions in L 1 R 4, it is enough to prove it for continuous initial data. By definition, the solutions of system 1.8 include classical transport and jumps on Σ. Let us first consider a point x 0, ξ 0 which is far from Σ, i.e. such that V x 0 ξ 0 0. Then, there exists a neighborhood Ω of x 0, ξ 0 and, τ > 0 such that for 0,, x, ξ Ω, t R, s ]t τ, t + τ [, g ±t, x, ξ = g ±s, Φ t+s ± x, ξ. As a consequence, t g ± + H ± g ± = 0 in Ω,

15 14 C. FERMANIAN KAMMERER AND F. MÉHATS and more generally, t g ± + H ± g ± = 0 for V x ξ 0. Consider now a point x 0, ξ 0 such that ξ 0 V x 0 = 0. Then, there exists a neighborhood Ω of x 0, ξ 0 and, τ > 0 such that for 0,, x, ξ Ω, t R and s t τ, t + τ the time τ corresponds to the length of an interval of time during which trajectories issued from points of Ω have at most one jump, g +t, x, ξ = g + s, Φ t+s + x, ξ 1 t+s τ+x,ξ<0t Φ τ+x,ξ + x, ξg t+s τ+x,ξ<0 T J + Φ τ+x,ξ + x, ξ g τ + x, ξ + t, J + Φ τ+x,ξ + x, ξ. We obtain.7 g+t, x, ξ = g+ s, Φ t+s + x, ξ [ T g +1 t+s τ+x,ξ<0 T g + τ + x, ξ + t, Φ τ+x,ξ + x, ξ τ + x, ξ + t, J + Φ τ+x,ξ τ + x, ξ + t, Φ τ+x,ξ + x, ξ + x, ξ ]. Similarly, we have.8 g t, x, ξ = g s, Φ t+s x, ξ [ +1 t+s τ x,ξ<0 T g+ τ x, ξ + t, J Φ τ x,ξ x, ξ ] T g τ x, ξ + t, Φ τ x,ξ x, ξ. Note that equations.7 and.8 are equivalent to the relation g t = L t s g s and give an explicit expression for the semigroup on a small time τ := t s during which the trajectories jump at most once. The result is then straightforward by Lemma.8. For the + mode the proof for the mode is similar, we have t +H + g+ [ = 1 τ+x,ξ+t 0H + 1τ+x,ξ 0 T g τ + x, ξ + t, J + Φ τ+x,ξ + x, ξ ] T g+ τ + x, ξ + t, Φ τ+x,ξ x, ξ. Observing that Φ τ+x,ξ + x, ξ = x, ξ for x, ξ Σ, we obtain for t 0, t + H + g + = Λ + x, ξδ Σ x, ξ [ T g + Σ,in T g Σ,in J + ] where we have used that for τ + x, ξ = 0 and t 0, we have 1 τ+x,ξ+t 0 = 1. This implies that g + satisfies Proof of Lemma.8. Let us begin with the first line. Recall that we have t + H ± Φ t ± = 0. Similarly, in view of τ ± Φ t ± x, ξ = τ ± x, ξ + t and writing Φ τ± ± x, ξ = Φ τ±x,ξ+t ± Φ t ± x, ξ,

16 KINETIC MODEL FOR GRAPHEN 15 we get t + H ± Φ τ± ± x, ξ = 0 and H ± Φ τ ± ± x, ξ = 0. Let us now prove the second line. Note that τ + x, ξ = 0 is an equation of the hypersurface Σ. For calculating H + 1τ+x,ξ<0, we write z = x, ξ, H + a = F z z a, where a is a test-function, and we observe that z F z = 0. Therefore, using that a is compactly supported, Green s formula reads F z z azdz = F z nz azdσ z F z azdz {τ +z<0} = Σ Σ F z nz azdσ {τ +z<0} where nz is the unitary exterior normal vector to Σ: nx, ξ = d V xξ + V x 1/ d V xξ V x Note that H + is transverse to Σ and points towards the region τ + < 0. Besides, one can check that H + x, ξ nx, ξ = Λ + x, ξ, where Λ + is defined by At this stage of the proof, we have obtained for any smooth compactly supported function a, H+ 1 τ+x,ξ<0 adxdξ = 1 τ+x,ξ<0h + a dxdξ = a H + ndσ Σ = Λ + a dσ, which gives the result..3. Mathematical justification of the approximation by the semi-group Strategy. In this section, we prove Proposition 1.7. The proof relies on a characterization of w t via pseudodifferential operators. Recall that if a C0 R, the semiclassical pseudodifferential operator of symbol a is defined by the Weyl quantization rule x + y op afx = π e iξ x y a, ξ fydydξ, f SR. This operator extends to functions f L R and one can prove that op a is a uniformly bounded family of operators of LL R since there exists a constant C > 0 and an integer N N such that.9 a C0 R 4, op a LL R C sup sup x β ax, ξ dx. β 3 ξ R R We refer to the books [1, 8, 34] for a complete study of pseudodifferential operators. The estimate.9 is not the standard Calderon-Vaillancourt estimate see [4] that is usually used. It has the advantage not to differentiate in the variable ξ and is inspired from [1] see also the survey []. A short proof is given in the Appendix Σ.

17 16 C. FERMANIAN KAMMERER AND F. MÉHATS for the convenience of the reader, we also recall the single symbolic calculus result that we shall use. Denote by C, the set of complex matrices and consider symbols a that are matrix-valued : a C 0 R, C,. Then, the operator op a is a matrix-valued operator acting on functions of L R, C. If w is the Wigner transform of the matrix density ϱ, we have the relation.10 a, w = tr op aϱ, where the bracket between the two matrices a = a i,j and w = wi,j is defined by.11 a, w = a i,j x, ξwj,ix, ξdxdξ. i,j R 4 We shall use this description of a, w in order to prove Proposition 1.7. Remark.9. The relations.10 and.9 imply that, under Assumption 1., the family w t >0 is a bounded family in the set of distributions. Since the initial density matrix ϱ 0 is supposed to be a Hilbert-Schmidt operator, there exists a sequence λ j j N of l N and a sequence of bounded normalized families ψ 0,j j N of L R such that As a consequence, for t R, ϱ 0 = j N λ j ψ 0,j ψ 0,j. ϱ t = j N λ j ψ j t ψ j t, where for any j N, the family ψ j t >0 is a family of solutions to the Dirac equation.1 i t ψ j = AD + V x ψ j, with initial data ψj t = ψ 0,j. Besides, the relation.10 yields a, w = tr op aϱ = j N λ j op aψ j t, ψ j t L R x. We denote by w j t the Wigner transform of the family ψ j t >0 which is defined by the relation a C 0 R 4, a, w jt = op aψ j t, ψ j t L R x. The Wigner function wj t is a by matrix and the bracket involved in the preceding relation is also the one defined in.11. In the following, we will characterize wj t in terms of the flow Lt,R. More precisely, we are going to prove that for any j N, wj t satisfies Proposition 1.7, which gives the result for w t. For this purpose, we use the results of [9, 17, 18, 19] which are stated for a Schrödinger equation with matrix-valued potential. This comes from the following observation : whenever V x = x, the operator AD + V x becomes a Schrödinger operator with a matrix-valued potential by taking the Fourier transform. As a consequence, the methods developed in [9, 17] for Schrödinger equation

18 KINETIC MODEL FOR GRAPHEN 17 with matrix-valued potential can be adapted to our setting. Furthermore, conical intersections have been classified in [6] and [15] and the Dirac-type equation 1.1, like the Schrödinger equations of [17, 18, 19], enters in the same class of crossings. Thus, it is not surprising that similar methods do apply. Note however that the jumps were omitted in [9] and [17]; as mentioned in [19], these jumps are required for the correctness of the proof of [17]. Then, the main steps of the proof will consist in : 1 The transport outside Σ. Localization in energy and use of space-time variables. 3 A normal form which reduces to a simple model called the Landau-Zener system. 4 The computation of the transitions on Σ that is performed via the normal form and the Landau-Zener system. In the following, we will take advantage of Remark 1.1 for taking into account only the jumps which occur inside the set U,R = {x, ξ R 4, ξ R }. Let us introduce the semi-group L,R which restricts the jumps to those occurring inside U,R, the semi-group L,R differs from L by exponentially small terms. It is this semi-group that we shall consider now. The real number R will be chosen as R = 1/8 see Notation 1. Let us now detail these steps. For simplicity, we omit the index j and simply consider a family ψ t >0, uniformly bounded in L R, of solutions to the Dirac equation.1 with initial data ψ 0 >0 and we denote by w t its Wigner transform at time t. We also denote by w ±t the scalar quantities w ±t = tr Π ± w t..3.. The transport outside the transition region. The analogue of Proposition.3 in [19] is the following Proposition.10. Let c Cc R 4, C, and let b C R, C with b compactly supported. If there exist C > 0 and s 0 > 0 such that r [ s 0, s 0 ] : Φ r ±suppc U,R = then for all χ Cc R, R and for all s [t s 0, t + s 0 ] tr χt cx, ξ b ξ R w ±t, x, ξ dx dξ dt = R d+1 tr χt cx, ξ b ξ w R ± s, Φ t+s ± x, ξ dx dξ dt R d+1 +O1/ R 5 + O1/R + O. This proposition is a refined version of the resolution of the kinetic system 1.4. Indeed, solving 1.4 in a subset of the domain { ξ > δ 0 } for some δ 0 on an interval of time [ s 0, s 0 ] such that the trajectories Φ s ± s [ s0,s 0] do not reach {ξ = 0} gives w ±t, x, ξ = w ±t s, Φ s ± x, ξ + O in D R 4. Proposition.10 authorizes to be at a distance of order OR of {ξ = 0}

19 18 C. FERMANIAN KAMMERER AND F. MÉHATS Since for scalar symbols a, we have d 1 dt aπ±, w [ t = op aπ ±, AD + V ] ψ t, ψ t i L R x, the proof of this proposition relies on a good understanding of the operator L = 1 [ op aπ ±, AD + V ]. i The main ingredients are the two following observations : For ax, ξ = cx, ξ b ξ R, the symbol ax, ξπ + ξ is smooth and we have.13 α, β N, C α,β > 0, x α β ξ ax, ξπ + ξ C R β, so that we can use the symbolic calculus theorems of the Appendix, paying attention to the rest terms. If B is an off-diagonal symbol, that is a symbol which satisfies the quantities B = Π + BΠ + Π BΠ +, χt op Bψ t, ψ t dt, which seems to be of order O1, can be proved to be of smaller order than expected by re-using the equation satisfied by ψ t. Proof of Proposition.10. Let us now focus on the proof itself. Using Proposition 4.1 and observing that Aξ = ξ Π + Π with Π + + Π = 1, we obtain L = op V x ξ a ξ ξ xaπ + + op B + OR + O, with B = a V Π + 1 ξ {aπ +, Π + } {Π +, aπ + } 1 ξ {aπ +, Π } {Π, aπ + } = a V Π + ξ {aπ +, Π + } {Π +, aπ + } = a V Π + + ξ Π + x a Π + + x a Π + Π + = a V Π + + ξ x a Π +. Here we have used Π + = Π + Π + + Π + Π +. As a consequence, B is an offdiagonal symbol. We write B = B 0 + B 1 with B 0 = a V Π +, we have.14 α, β N d, C α,β > 0, x, ξ R 4, β ξ α x B j Cα,β R β 1+j. The result comes from the next lemma which concludes the proof. Lemma.11. For any χ C 0 R, we have χt op B 1 ψ t, ψ t = OR + O, χt op B 0 ψ t, ψ t = OR 5 1/ + O.

20 KINETIC MODEL FOR GRAPHEN 19 Proof. We begin with B 1. Since B 1 is off-diagonal, we can write B 1 = [Π B 1 Π + Π + B 1 Π ξ 1, Aξ] = [Π B 1 Π + Π + B 1 Π ξ 1, τ + V x + Aξ]. After quantization, we get [ op B 1 = op Π B 1 Π + Π + B 1 Π ξ 1, ] i t + V x + AD +OR + O /R. Once applied to ψ which satisfies the Dirac equation.1, we obtain the announced relation. Note that we have obtained more generally that if B j is off-diagonal and satisfies the relation.14, then.15 χt op B j ψ t, ψ t = O R 1+j R + /R = O R 1+j R + In particular, for B 0, we obtain χt op B 0 ψ t, ψ t = OR 3 1/ + O /R that we want to improve. Therefore, we go one step further in the symbolic calculus and we write [ op B 0 = op Π B 0 Π + Π + B 0 Π ξ 1, ] i t + V x + AD + i op { Π B 0 Π + Π + B 0 Π ξ 1, τ + V x } + i op { Π B 0 Π + Π + B 0 Π ξ 1, Aξ } { Aξ, Π B 0 Π + Π + B 0 Π ξ 1} +OR 4 + O /R 3. Paying attention to all these terms, we observe that [ op B 0 = op Π B 0 Π + Π + B 0 Π ξ 1, ] i t + V x + AD The matrix i op V ξ Π B 0 Π + Π + B 0 Π ξ 1 + OR + O. B = V ξ Π B 0 Π + Π + B 0 Π ξ 1 satisfies.14 with j = and we claim that B is also off-diagonal. consequence, equation.15 gives χt op B ψ t, ψ t dt = OR 5 3/ + OR 3 1/, which concludes the proof of Lemma.11. It remains to prove the claim, a simple calculus shows that As a B = 1 ξ 3 V ξπ B 0 Π + Π + B 0 Π + ξ 1 V ξ Π B 0 Π + Π + B 0 Π.

21 0 C. FERMANIAN KAMMERER AND F. MÉHATS Therefore, Π ± B Π ± = ξ 1 Π ± V Π B 0 Π + + Π B 0 V Π + Π ± ξ 1 Π ± V Π + B 0 Π + Π + B 0 V Π Π ± = ξ 1 Π ± V Π + B 0 Π + + Π + B 0 V Π + Π ± = ξ 1 Π ± [ B 0, V Π +] Π ± = 0 since B 0 = a V Π +, which proves that B is off-diagonal. Notation 1. In the following, it will be convenient to denote by η any rest term smaller than O1/R 5 +O1/R +O +OR 3. The term in R 3 will be useful in the following. Note that when R = 1/8, we have η = O 1/ Localization in energy. The memory of the mode by use of a matrix-valued symbol of the form aπ + or aπ, with a scalar, can be replaced by a localization in energy. This requires to work in space time variables and has the advantage that we are reduced to use scalar symbols. Using scalar symbols will be convenient in the next section when we will perform a normal form and use a Fourier Integral Operator. The energy surfaces of the space-time phase space R 6 t,x,τ,ξ are the sets.16 E ± = {t, x, τ, ξ R 6, τ = ξ V x}. Recall that the dual variable of the time t is interpreted as an energy τ. In the following, we shall use semi-classical pseudo differential operators with symbols depending on the variable t, x, τ, ξ R 6 with the choice of the Weyl quantization in the time variables, as it was already the case for the space variables. The localization in energy is done by use of a cut-off function θ C 0 R such that 0 θ 1, θτ = 0 for τ > 1 and θτ = 1 for τ < 1/. This function θ is fixed from now on. Lemma.1. Let a C0 R 4 and set c,r x, ξ = ax, ξ1 θξ/r, then for all χ C0 R, χt op c,r Π ± ψ t, ψ t dt = L R O1/R + O R τ ± ξ + V x + op χtc,r x, ξθ R ψ, ψ L R 3 t,x. Remark.13. It is because the localization in energy is made in balls of size that we need to perform the jumps: they guarantee that the energy of the created trajectory do not differ at order O but at least at order O. Remark.14. Note that the presence of the eigenprojector in the symbol induces restriction on both components of the function ψ t. Indeed, by the symbolic calculus of the Appendix and by equation.13 we have χt op c,r Π ± ψ t, ψ t L R dt R = χt op c,r Π ± ψ t, Π ± ψ t L R dt + OR. R

22 KINETIC MODEL FOR GRAPHEN 1 Proof of Lemma.1. We set θ ±,R x, τ, ξ = θ τ ± ξ + V x R Following the lines of the proof of Lemma 5.1 in [18], we observe that since 1 θ vanishes identically close to 0, one can write 1 θ + 1 τ + ξ + V x,r x, τ, ξ = R τ + ξ + V xg R for some smooth function G, with. τ + ξ + V xπ + ξ = Π + ξτ + Aξ + V x. Therefore, we can use the equation satisfied by ψ t, symbolic calculus and the estimate.13 to obtain op χc,r Π + ψ, ψ = op χc,r θ +,R Π+ ψ, ψ + O R + O. It remains to get rid of the matrix Π + ξ. In view of χc,r θ +,R = χc,rθ +,R Π+ + χc,r θ +,R Π, we only need to prove that op χc,r θ +,RΠ ψ, ψ = Oη. We observe that θ +,R c,r = θ +,R 1 θ,r c,r, and 1 θ 1 τ ξ + V x,r Π = R G R τ ξ + V xπ 1 τ ξ + V x = R G R Π τ + Aξ + V x. By using again the equation, symbolic calculus and estimate.13, we can write op χa + θ +,R Π ψ, ψ = op χa + θ +,R 1 θ,r Π ψ, ψ + O R + O = Oη. The proof for the minus mode is similar The normal form. For computing the transitions, we use a normal form result. For this, we need to work microlocally in space-time variables. Following [6, 14, 13], close to a point t 0, x 0, ξ 0 = 0, τ 0 = V x 0, there exist a change of coordinates κ : s, z, σ, ζ t, x, τ, ξ with s, σ R and z = z 1, z, ζ = ζ 1, ζ R, and a matrix B such that.17 τ + V + Aξ κ = t B σ + Ãs, z 1 B, with s z1 Ãs, z 1 =. z 1 s Moreover, this change of coordinates preserves the symplectic structure of the phase space R 3 t,x R 3 τ,ξ : the variables σ and ζ are respectively the dual variables of s and z. Besides, in view of Section 6. of [14], there exists a function γ > 0 such that t BB = γid.

23 C. FERMANIAN KAMMERER AND F. MÉHATS The construction of the canonical transform is based on the vectors H and H defined in.1 and. see [6] and the analysis performed in [17, 13]. The variable s is chosen such that the trajectories which reach {ξ = 0} are included in {s < 0} and those which leave ξ = 0 are included in {s > 0}. Besides, one extends the vectors H and H as vectors of T R 3 t,x by adding the coordinate 1 along t and the coordinate 0 along τ and we keep calling them H and H. The resulting vectors are the limit on {ξ = 0} and along the flows of the Hamiltonian vector fields associated with the functions τ + V x ± ξ. They are sent by dκ on the limit on {s = z 1 = 0} and along the flow of the Hamiltonian vector fields associated with γ σ± s + z1. A simple calculus shows that since the canonic symplectic form ω is preserved by canonical transform, the relation.3 implies that H is sent on γ s + σ, the limit as s goes to 0 of the Hamiltonian field associated with γ σ s + z1 and H is sent on on γ s σ, the limit as s goes to 0 of the Hamiltonian field associated with γ σ + s + z1. This observation allows to relate the modes after the change of coordinates. As a consequence, in these new variables s, z, σ, ζ, the geometry of the crossing is simple and we have.18 S := {ξ = 0, τ + V x = 0} = κ {s = 0, z 1 = 0, σ = 0}, E ± = κ { σ s + z1 = 0}, where the energy sets E ± are defined by.16. Finally, in the construction of the canonical form κ, the function z 1 x, ξ can be related to the variables x and ξ according to.19 z 1 x, ξ = ξ V x V x 3/ + O ξ. Similar formula can be written for the functions s and σ. However, in the sequel, we will only use the formula for z 1. Then, thanks to Theorem 3 of [6], it is possible to pass equation.17 at the quantum level : there exists a unitary operator K and a matrix B 1 such that K op t B op σ + Ãs, z 1 op B K = op τ + V x + Aξ + O, where B = B+B 1. The operator K is a Fourier Integral Operator associated with the canonical transform κ see [8] or [14]. It allows to pass at the quantum level the relation.17 induced by the change of variables κ. An important property of these Fourier Integral Operators is that they are compatible with pseudo differential calculus in the sense that for all a C 0 R 6, K op a κ 1 K = op a + ON a, where N a = sup α K, β, α + β sup τ,ξ R 3 R 3 α t,x β τ,ξ at, x, τ, ξdt dx

24 KINETIC MODEL FOR GRAPHEN 3 for some K N. In particular, when one applies this relation to a symbol of the form τ ± ξ + V x a,r x, ξ = χtc,r x, ξθ R Π ± ξ, one gets N a,r = OR + O1, and N a,r = Oη. We will use this property to translate the quantities that we want to study in the variables t, x, τ, ξ in these new variables s, z, σ, ζ. More precisely, we set v = op B K ψ, then v solves microlocally in L R 3 s,z the system.0 i sv = Ãs, z 1v + O and op t BaB κψ, ψ = op L R av, v 3 t,x L R 3 s,z + Oη where η denotes a rest term as defined in Notation 1. In particular, for scalar functions a, we have op γa κψ, ψ L R 3 t,x = op av, v L R 3 s,z + Oη In what follows, we shall focus on the analysis of this family v. Let us now write the Markov process L,R in the new coordinates, we shall denote by L,R the resulting semi-group. The hopping region is a neighborhood of size R of the intersection of the energy sets E + and E, therefore it can be replaced by Ũ,R = { z 1 + s C 1 R } for some constant C 1 > 0. As already noticed before, turning C 1 into some C > 0 will only induce exponentially small additional transitions, so that one can take any neighborhood of size R of the set S. As we have already observed, by the geometric properties of canonical transforms, the Hamiltonian trajectories of our system are preserved by κ and one is able to identify each branch of the trajectories: the trajectories for the plus mode are Hamiltonian trajectories of σ s + z1 and the trajectories for the minus mode are those of the Hamiltonian σ + s + z1. We denote by Φ ± these trajectories and we observe that they write.1 Φℵ ± s, z, σ, ζ = s ℵ, z, σ ±s, ℵ z 1, σ, ζ 1 ℵ ±s, z 1, σ, ζ, where we set ζ = ζ 1, ζ and σ ±s, ℵ z 1, σ = σ s ℵ + z1 ± s + z1 by the conservation of the energy and ζ 1 ℵ ±s, z 1, σ = ζ 1 + Oℵ. The transitions occur when the gap is minimal along the trajectories, that is when s = 0. Besides, when the transitions occur, one has ξ V x = 0, which implies ξ V x = ξ V x and the relation.19 gives T x, ξ = exp π ξ V x V x 3 = T LZ z1 1 + OR 3

25 4 C. FERMANIAN KAMMERER AND F. MÉHATS where T LZ η = e πη. Since T x, ξ and T LZ z 1 / differ of a term of order η, we define the flow L,R with the transition rate T LZ z 1 /. Finally, we observe that the drift is made in the direction of H + H. By the description above, dκ sends H + H on a vector collinear to σ. As a consequence, we deduce that there exists a map σ δσ such that. J± := κ 1 J ± κ 0, z, σ, ζ = 0, z, σ + δσ, ζ. Using.18, we deduce δσ = z 1. Let us now reformulate our problem in these new variables. Recall that we work in the region Ũ,R. Let b ± s, z, ζ be two smooth functions compactly supported in {s > 0} and such that the trajectories reaching their support have only experienced one transition during an interval of time of length ℵ. We also suppose that the functions b ± s + ℵ, z, ζ are supported in {s < 0}. We consider the symbol c out c +,out,r, c,out,r defined by c ±,out,r s, z, σ, ζ = b± s, z, ζθ λ± s, z 1, σ R where θ is the cut-off function of Lemma.1 and λ ± s, z 1, σ is the energy λ ± s, z 1, σ = σ z1 + s.,r = Note that the localization in energy and the fact that we work in the hopping region yield that σ = OR in the zone of interest; for this reason we do not need to assume that b ± depends on the variable σ. We now want to compute L ℵ,R cout,r the pull back by the semi-group L,R in the normal coordinates. The observable c out,r has two parts c+,out,r and we have to consider the random trajectories that reach the support of each of these and c,out,r functions. More precisely, for c +,out,r, we consider the plus trajectories that reach its support; however, these trajectories may have known a jump and either they result from plus trajectories, either they result for minus trajectory. Similar description holds for trajectories reaching the support of c,out,r At that point of the analysis, we notice that by point 3 of Assumptions, in the ingoing region, one of the mode is negligible. Without loss of generality, we can assume that the contribution of trajectories which arise from the minus mode is negligible. For summarizing, the picture is the following : for calculating the backward image by the semigroup of c +,out, we only,r need to consider the plus trajectories which reach its support, for calculating the backward image by the semigroup of c,out,r, we need to consider the minus trajectories which reach its support and these trajectories arises from plus trajectories which have had a jump. We denote by Φ s ± the Hamiltonian trajectories associated with λ ± and we observe that along a trajectory, the variable ζ 1 is constant up to OR term, since ζ 1 = O1 with observation time of order R. Let us now calculate c +,out,r. We observe that for s, z, σ, ζ = ζ 1, ζ supp c +,out we have s, z, σ, ζ1 + OR, ζ = Φ ℵ + s ℵ, z, σ, ζ.,r,